# Help Solving Two Proofs - Tan X + Cot X = (Sec X)(Csc X)

• chase222
Now, think about the Pythagorean identity sin^2 x + cos^2 x = 1. What do you get if you subtract cos^2 x from both sides? For the first one, you have to recognize that tan x = sin x/cos x and cot x = cos x/sin x, so tan x + cot x = (sin x/cos x + cos x/sin x). Can you put that into a single fraction?
chase222
I need help solving 2 proofs:

tan x + cot x = (sec x)(csc x)

I changed the left side to:

tan x + 1/tan x = (sec x)(csc x)

then crossed out the tan:

1 = (sec x)(csc x), but I got stuck there.

The next one I had trouble with was:

tan^2 x - sin^2 x = (tan^2 x)(sin^2 x)

I saw the left side being a^2 - b^2, so I factored it into:

(tan x + sin x)(tan x - sin x) = (tan^2 x)(sin^2 x)

I then changed the tan into sin/cos:

((sin x/cos x) + sin x)) ((sin x/cos x) - sin x)) , but got stuck there.

Can you help me solve these proofs?

Try to write everything in terms of sine & cosine...You'll get them easily.

Daniel.

So for the second one:

tan^2 x - sin^2 x = (tan^2 x)(sin^2 x)

(sin^2 x/cos^2 x) - sin^2 X = (sin^2 x/cos^2 x)(sin^2x)

So on both sides so the sin^2 x cancel, leaving it like:

cos^2 x = cos^2 x?

And for the first one:

tan x + cot x = (sec x)(csc x)

I changed it to:

sin x/cos x + 1/(sin x/cos x) = (1/cos x)(1/sin x)

What would I do from here?

Bring it to the same denominator (in the LHS) and after simplifying the denominators,u'll find

$$\sin^{2}x+\cos^{2}x =1$$

Daniel.

chase222 said:
I changed the left side to:

tan x + 1/tan x = (sec x)(csc x)

then crossed out the tan:

1 = (sec x)(csc x), but I got stuck there.

"crossed out the tan" is not a mathematics term! I'm serious- think about exactly what you are doing there. tan x+ 1/tan x is NOT equal to 1 for all x!

chase222 said:
(sin^2 x/cos^2 x) - sin^2 X = (sin^2 x/cos^2 x)(sin^2x)

So on both sides so the sin^2 x cancel, leaving it like:

cos^2 x = cos^2 x?

Okay, sin^2 x/cos^2 x) - sin^2 x= (sin^2 x)((1/cos^2x) - 1) so canceling sin^2 x leaves (1/cos^2 x)- 1 = sin^2 x/cos^2 x

That is NOT "cos^2 x= cos^2 x" but if you multiply both sides by cos^2 x you get something almost as easy.

1.

## What does the equation Tan X + Cot X = (Sec X)(Csc X) represent?

The equation represents a trigonometric identity, which is a mathematical statement that shows the relationship between different trigonometric functions.

2.

## How do I solve this equation using trigonometric identities?

To solve this equation, you can use the Pythagorean identity, which states that sin^2(x) + cos^2(x) = 1. You can also use the reciprocal identities, which state that tan(x) = sin(x)/cos(x), cot(x) = cos(x)/sin(x), sec(x) = 1/cos(x), and csc(x) = 1/sin(x).

3.

## What are the steps to solving this equation?

The first step is to rewrite the equation using the reciprocal identities. Then, use the Pythagorean identity to simplify the equation. From there, you can use algebraic manipulation to solve for x.

4.

## What is the purpose of solving this equation?

Solving this equation can help you better understand the relationship between different trigonometric functions and how they can be manipulated using trigonometric identities. It can also be used to solve more complex trigonometric equations and problems.

5.

## Are there any tips for solving this equation?

It can be helpful to draw a right triangle and label the sides and angles with the corresponding trigonometric functions. This can provide a visual representation of the equation and help with understanding and solving it. Additionally, make sure to check your work and simplify the equation as much as possible to ensure accuracy.

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